PFC/JA-82-23
Three-Dimensional Theory of Waveguide-Plasma Coupling
A. Bers, K.S. Theilhaber
Plasma Fusion Center
Massachusetts Institute of Technology,
Cambridge, MA 02139
Tc/JF-2 -Z 3
THREE-DIMENSIONAL THEORY
OF WAVEGUIDE-PLASMA COUPLING*
A. BERS, K.S. THEILHABER
Plasma Fusion Center,
Massachusetts Institute of Technology,
Cambridge, Massachusetts,
United States of America
ABSTRACT. A general, linear theory of coupling electromagnetic fields from free-space waveguide arrays
to an inhomogeneous plasma in a magnetic field is presented. In contrast to previous analyses, which assumed
parallel-plate waveguides, waveguides having closed cross-sections of arbitrary shape are considered; full account
is taken of all the waveguide modes. Far away from the coupling region, the plasma is assumed to be absorptive
and the waveguides to propagate only their dominant modes. The formulation is geared to obtain the reflection
coefficients in the waveguides and the excitation of waves in the plasma. The results are applicable to RF heating
of plasmas whose size is large compared to the waveguide array, as would be the case in a fusion reactor, and
valid for the frequency regimes of either ion-cyclotron (harmonic), or lower-hybrid, or electron-cyclotron
(harmonic) heating.
In what is generically known as 'RF heating and
current generation' of plasmas, electromagnetic (EM)
energy from high-power sources external to the plasma
is used to excite electromagnetic fields that propagate
into the plasma and eventually dissipate their energy
and momentum on the charged particles of the plasma,
thus achieving the desired heating and/or current
generation. One important aspect of such interactions
is the coupling of power from the EM sources in freespace to the EM fields in the plasma. In magnetically
confined plasmas with appreciable bulk temperatures,
which are of interest in achieving fusion temperatures
in a quasi steady-state, coupling structures cannot be
inserted into the plasma and such coupling must take
place at the plasma edge. Under these circumstances,
and for technological reasons which we shall not go
into here, it is useful to consider EM sources at high
frequencies whose energy can be brought conveniently
to the plasma edge by waveguides. Specific heating
and current generation schemes require that the excited
fields have a prescribed spatial spectrum. This then
requires that the waveguides be fed with specific relative amplitudes and phases, thus forming waveguide
arrays.
In this paper, we formulate a general linear theory to
describe the coupling of EM power from waveguides
to a plasma in a magnetic field. The coupling region
(Fig. 1) is taken as the inhomogeneous plasma extending
from the free-space- waveguide openings- in the-plasmawall to where the desired plasma modes are excited.
The excited plasma modes are assumed to be dissipated
beyond the coupling region, further into the plasma
core. The plasma is taken to be of sufficiently large
(reactor) size so that the limited extent of the coupling
region can be modelled in slab geometry (Figs 2
and 3). For definiteness, the plasma in the coupling
region is described by the cold-plasma model with
inhomogeneous density and magnetic field in one
direction, into the plasma. The walls of the plasma and
waveguides are assumed perfectly conducting, and the
waveguides are allowed to be of arbitrarycross-section;
the medium in the waveguides is taken as free-space.
Far from the plasma wall the waveguides are assumed
to propagate only their dominant mode. Given the
amplitudes and phases of the incident fields in each of
the waveguides and the unperturbed characteristics
of the plasma in the coupling region, we show how to
determine the reflection coefficient in each of the
waveguides and the excited fields and power flow into
the plasma modes in and beyond the coupling region.
The plasma modes amenable to waveguide excitation
for heating a reactor-type plasma fall into three fre-
* Work supported by DoE Contract DE-AC02-78ET-51013
and NSF Grant No.ENG 79-07047.
quency ranges: (a) ion-cyclotron range of frequencies
(ICRF), from the ion-cyclotron frequency (2/27r) to
1. INTRODUCTION
NUCLEAR FUSION, Vol.23, No.1 (1983)
41I
BERS and THEILHABER
PLASMA
WALL
WAVEGUIDE
ARRAY
CENTRAL
PLASMA
EM
SOURCI ES
EDGE
PLASMA
II
n*
~TO
COUPLING
REGION
FIG.1. Waveguide-plasma couplinggeometry. The coupling
region extends from the plasma wall into the edge plasma;
it does not include the hot, centralplasma where the excited
waves are assumed to be dissipated. For a large plasma, the
coupling region can be modelled in slab geometry.
PERFECT CONDUCTOR
WALL
FREE -SPACE
PLASMA
-------- ---
X
yb
FIG.2. Geometry used in the analysis of coupling from a
single waveguide to a plasma.
a few times (92i/27r); (b) lower-hybrid range of frequencies (LHRF) which is around and above the ion
plasma frequency (copi/21r). (c) electron cyclotron range
of frequencies (ECRF) from the the electron cyclotron
frequency ( 2 e/27r) to a few times ( 2eI/27r). For magnetically confined fusion plasmas, in externally applied
magnetic fields in the range of 4-6 T, the ICRF would
involve waveguides whose cross-sectional dimensions
would be comparable to a reactor plasma radius; for
plasma densities in the range of 1020 - 1021 mn3 , the
LHRF fall in the usual microwave regime and the
42
waveguide dimensions become smaller than a reactor
plasma radius; finally, in the ECRF the waveguides are
much smaller than the plasma radius. In all cases, the
finite size of all the cross-sectional dimensions of the
waveguide relative to the plasma dimensions must be
accounted for in the coupling problem.
Previous analyses of the waveguide-plasma coupling
problem have concentrated on the LHRF [1-4]. All
these analyses were carried out in detail in only two
dimensions: the waveguides were assumed to be
perfectly conducting parallel-plates of infinite extent
in the direction perpendicular to the confining magnetic
field B 0 , and the excited plasma modes were thus
forced to have no variation in that direction. Most
analyses were also concerned mainly with the dominant
mode in the waveguides and neglected all of the higherorder modes that are required for a proper description
of the fields at the plasma wall where the waveguide
ends. In the present analysis, these restrictions are
removed; full account is taken of the finite dimensions
of the waveguides in both the direction along B0 and
the direction perpendicular to Bo, and all the higherorder waveguide modes are accounted for in the vicinity
of the waveguide opening in the plasma wall. In
addition, we do not restrict ourselves to any particular
frequency regime. Instead, the differential equations
describing the fields in the inhomogeneous plasma, in
the coupling region, are derived to apply to any
of the frequency regimes. In general, these are four
coupled first-order differential equations with nonconstant coefficients, and without further approximations their solutions must be arrived at by
numerical techniques. Such numerical integrations are
not carried out in this paper, but as a guide to work
PERFECT
CONDUCTOR
WALL
.----,----- ----------x
------1-- - ------X
FREE-SPACE
WEGUIDE
vno,vBo
/PLASMA
FIG.3. Geometry used in the analysis of couplingfrom multiple
waveguides - waveguide arrays - to a plasma.
NUCLEAR FUSION, vol.23, No.1 (1983)
WAVEGUIDE-PLASMA COUPLING
in the future some approximate solutions are outli
This together with the mode expansion for the fie
in the waveguides is shown to lead to a complete
solution of the linear coupling problem.
Section 2 describes the mode expansion of the'
guide fields. Section 3 describes the plasma fields
their Fourier transforms in the two directions of
assumed plasma homogeneity. Section 4 gives the
solution of the coupling problem from a single wa
guide. Section 5 generalizes the solution of the co
problem to multiple waveguides (waveguide arrays
independently fed in amplitude and phase. Sectio
gives some approximate solutions relevant to the I
Section 7 summarizes the results and discusses the
limitations of the assumed model and analysis.
The dominant mode is characterized by wavenumber
kd = (27r/Xg), normalized transverse field patterns ed
and hd, waveguide admittance Yd, incident-field
amplitude V., (referenced to x =0, or any place an
integer multiple of Xg/2 to the left of x = 0), and
complex reflection coefficient R. The higher-order
modes are all evanescent from x =0 toward x <0
with spatial decay rate aH and field amplitude KVH;
the summation subscript H stands for all the TE and
TM modes in this category [5].
3. THE PLASMA FIELDS
2. THE WAVEGUIDE FIELDS
We consider an arbitrary cylindrical waveguide,
one which is uniform in the x-direction and of arbi
closed cross-section transverse to x. Figure 2 illust
a waveguide of rectangular cross-section, but our
description is equally applicable to any other cross
section, e.g. circular or elliptical. For simplicity, mA
shall assume that the waveguide walls are perfectly
conducting, and that the plasma does not penetrat,
into the waveguide so that the medium enclosed b,
waveguide walls is taken as free-space (eo, Ao). Th
solution of Maxwell's equations in such cylindrical
waveguides is well known [5 ]. The electromagneti,
fields can be expressed as the superposition of an
infinite set of orthogonal E (or TM) modes and H
(or TE) modes which are complete. For our purpc
we assume that, at the frequency of interest, the
waveguide propagates only its dominant (lowest-or
mode, and all other (higher-order) modes are belo
cut-off. The transverse fields in the waveguide can
then be written as follows:
-W
V. 0
E=+R
+
e
ikdx
+ Re
VH ea
-ikdx)
(1)
H
W
w
YdVo
( ikdx
l+R \e
+
-
-ikdx
-Re
X
YHVH eaHxH(y, z)
H
NUCLEAR FUSION, vol.23, No.1 (1983)
n KX
d
-Ydt ny-YKX
1
y
(2)
Y+
i
ny nz
/
i X I-ZoXY
O +
2
\
/oxZZo~
(3)
dP
= - n
-KX
--n,
-i-
ZK
Y
KL
Zoy- i
n nz
ZoZ
(4)
d--
dZolz
d
ed (y,z)
H(y, Z)
In the coupling region, near the waveguide openings
in the perfectly conducting wall (Figs 2 and 3), at a
given radian frequency w, the plasma will be assumed
to be described by its cold, inhomogeneous dielectric
tensor. For simplicity, the inhomogeneity in the
unperturbed density no and applied magnetic field
Bo= iBo will be taken in the x-direction. The electromagnetic fields in the plasma can then be Fourieranalysed in y and z, and the complete solution for the
fields, through Maxwell's equations, is determined by
the solution of four coupled first-order, ordinary
differential equations (o.d.e.'s) in x, representing the
coupling of 'slow' and 'fast' waves [6, 7]:
+i(n2-K
=-innz
_=i (n 2-Ki+
+±inynz&z- k-
K_
KX
(5)
&y
oy+-X-n
oKi-yoCKn
Iz
(6)
where we normalized co-ordinate distances to (c/o),
e.g. (ox/c) =, and take the Fourier transform of the
fieldsy and z: (Qnynz)- and A($,ny,nz)**l,
where ny =(cky/w) and nz = (ckz/w) are the indices of
refraction in y and z, respectively, and ky and kz are
the wavenumbers in these directions. In the above
43
BERS and THEILHABER
equations Ki, Kx and K11, the cold-plasma dielectric
tensor elements, are functions of , and ZO=\/r0eO.
These equations are all first-order and are well suited
for a numerical treatment. The only singularity in the
coefficients occurs when K_( - 0, which is simply the
hybrid resonance (lower or upper hybrid). If we
impose the values of &y and Fz at t =0, and require
that only outgoing or evanescent waves exist as t -+ -,
then these boundary conditions determine completely
the solution of Eqs (3-6). Without further approximations, these equations must be solved numerically
as follows. At the end of the coupling region into the
plasma, the 'fast-'and 'slow'-wave fields will be assumed
uncoupled and described by their WKB form. Furthermore, assuming that the plasma beyond the coupling
region is absorptive, we can take the WKB form for
each as outgoing waves of, say, unit amplitudes. The
'differential equations can then be integrated backwards in x to obtain &To - 9( =0, ky, k,) and
tTo ;T ( =0, ky, k,) for all ky and kz of interest.
The integration of the coupled differential equations
for 9T and RT must be carried out with care so that
resonant (absorptive) or cut-off (evanescent) regions
are properly taken into account for each pair of ky
and k,. Further details of this are discussed in Section 7.
The integration of the coupled differential equations
can be considered to determine the matrix (Q) that
gives
(
(7)
'
where Yp= Yp,(,,ky, kz) is the plasma admittance
matrix. The electromagnetic fields transverse to x in
the plasma at x =0 are then obtained by computing
the inverse Fourier transform of T(T = 0) = 9To and
N'1(1=0) rT
P
0>)
-P
HT(x=
0
)=
(11)
and
(12)
fT(x=0)= H (x=0)
Using Eqs (1), (2), (8), (9), and (10), we can write
expressions (11) and (12) as
V+oe
2
dVkT
HLVHeHJ (2ir) 2
+
T e
H
w
d
-R
I+R
d
+0
rT
Toe
H
(13)
-
H
Id
2
kT =*
2
S2i)
I
irT
(14)
poTo e
where we note that
w IR
d 1 +-R
Pw
(15)
d
is the admittance of the plasma presented to the
dominant mode of the waveguide. Equations (13) and
(14) determine, in principle, this admittance (and
hence R) as well as the amplitudes of the higher-order
modes per unit of incident-field amplitude, i.e.
(VH /V10 ) -VH . To write this out explicitly, we
take the Fourier transform in y and z of Eq.(13) to
obtain
(16)
eTo - V+o ed + YV-HFH
H
where ei= f d2rTei exp (- iST -?T) is the Fourier trans2
9T(x
9W(x=0)4(x=0)
d kT2 - irT
(2,T) gToe
-r
(8)
form of
to find
1,
and substitute Eq.(16) into expression (14)
2
(21r)2
Toe
(9)2
(9
Yd
hd~
YHV-HhH
H
where kT =
yky+ Ak and rT = yy + iz, and, using
Eq.(7),
Sj7TO - Yp(
d2 kT
0)-&T =1Ypo - dT
(10)
(0)
4. THE WAVEGUIDE-PLASMA COUPLING
For coupling with a single waveguide, as shown in
Fig.2, the waveguide transverse electric and magnetic
fields at x = 0 must match onto the plasma fields, i.e.
44
(
2
-#
po
+
1
i
-H H
d+
I
r
r
(17)
H
Now, using the orthogonality properties of the waveguide field patterns [5], and introducing the coupling
admittances
=kT
r-dik
(21r)2
YPp'j
(18)
NUCLEAR FUSION, Voi.23, No.1 (1983)
WAVEGUIDE-PLASMA COUPLING
Ni= f d2rThi exp (-ik rT) in the Fourier t ransform of hi, we extract from Eq.(17)
where
-o= V.0[,
(25)
Hf
H
(19)
V+of~±
Pw
YdH H Ydd
Yd
9-H
+
To
H
Y
Y1
V.HEH}
d
YH
-H
(26)
j
H
and
(YHH'H +YH H)-H
Using these we can then obtain the electromagnetic
field spectra in the plasma as excited by the dominantmode incident-field amplitude (V. 0 ) in the waveguide,
and the power flow into the plasma.
YH'd
H
Equations (19) and (20) can be solved for the
normalized higher-order mode field amplitudes K-H
PW
and Yd which by relation (12) determine the
complex reflection coefficient R:
R=
~PW
I-$
To have more control on the excited spectrum of
1 +yPw
(21)
Yd
where
PW
Pw_ Yd
d - YW
Note that Eq.(19) represents the coupling of the
dominant mode to the higher-order modes, and
Eq.(20) gives the coupling of the higher-order modes
among -themselves.
The importance of higher-order modes can be
determined by solving various truncated forms of
Eqs (19) and (20). If we ignore all the higher-order
modes, then from Eq.(19) we simply have
PW
Yd =
fields in the plasma it is common to use multiple
waveguides appropriately phased, i.e. waveguide arrays,
as shown for example in Fig.3. The coupling analysis
of the preceding section is easily generalized to this
case. The matching of the tangential rand H fields,
Eqs (11) and (12), take on the more general form
Yid
Ydd -~ Yd
V-H H
V+
VPw
lY
ed ,
f
d
Y,
w0 d
VH H
H
To eikT-rT
1
(28)
Po0&To e
T2
--
2
H
w
(22)
dd
If account is taken of one higher-order mode, say H = 1,
Eqs (19) and (20) give a two-by-two matrix of equations
which solves readily to give
PW=
Yd
5. COUPLING FROM WAVEGUIDE ARRAYS
TO THE PLASMA
where the summation over the superscript w is over all
the waveguides in the array. The Fourier transform of
Eq.(27) gives
(23)
-+y
w
[
and
(29)
Vw
w
H
-Yd
Y1 + Y11
(24)
In general, Eqs (19) and (20) form a square matrix of
equations of order equal to the number of higher-order
modes plus one. For computational purposes proper
ordering and truncation can be determined from
estimates of expression (18) for different pairs of modes.
Having determined Ydw and the V-H, Eqs (13) and
(14) give
NUCLEAR FUSION, Vol.23, No.1 (1983)
where " (ky, kz) -+e (y, z) is a Fourier transform pair,
and using expression (29) in Eq.(28) we obtain
w
ZPW W~W
C_
W W _
H
d2 kT
2
= (2r)
w
4
po'
W
Vwe d+
w
H
ikTrT
V.-H H e
(30)
45
BERS and THEILHABER
Let the waveguides be designated by superscripts
w = a, ,-..., and the modes in each waveguide by the
vertically aligned subscript (i, j)= d or H. Using the
orthogonality properties [5] of the a-waveguide field
patterns, and introducing the coupling admittances
d2 kT
(2ir)2 T
yw
U
_.a
where in (ky , k,)
Yp -W
(31)
(1
3
-
hi (y, z) is a Fourier transform pair,
we find from Eq.(30):
YPa
waaaa
Wa
H
wia
H
Sa
(32)
w
wka
guide. Note from (34) that the normalized, complex
amplitudes V. contain the relative phasings in the
incident dominant-mode fields of the various
waveguides.
Proceeding in similar manner to apply to Eq.(30)
the orthogonality properties of the field patterns of
the other waveguides 0, y, ... we obtain a set of
Eqs (32) and (33) for each waveguide by simply
replacing a with #, y, ... . The simultaneous solution
of all these sets of equations gives the higher-order
mode amplitudes and reflection coefficient (through
Pw
Yw and (21)) in each of the waveguides. Again, proper
ordering and truncation of all these equations can be
arrived at from estimates of the coupling admittances
(31). Thus using the results in expression (29), we
can finally also find the electromagnetic field spectra
in the plasma, as excited by a given set of dominantmode incident-field, complex amplitudes (V0) in the
waveguide array, and the power flow into the plasma.
and
( -H
'H
H
H H
-H
6. COUPLING AT
LOWER-HYBRID FREQUENCIES
WHa
YHd
(33)
YH d +0
woa
where the mode amplitudes have been normalized to
the incident dominant mode amplitude of waveguide a,
+0
+0
-H
Va ;and VH_
+0
+0
(34)
The first two terms on the left-hand sides of Eqs (32)
and (33) and the first terms on their right-hand sides
are, of course, the same as in Eqs (19) and (20),
respectively, and represent the coupling of all the
modes in waveguide a. The last term on the left-hand
side of Eq.(32) represents the coupling of the
dominant mode in waveguide a to the higher-order
modes in all the other waveguides, and the last term
on the left-hand side of Eq.(33) is due to the coupling
of the higher-order modes in waveguide a to the higherorder modes in all the other waveguides. Finally, the
second terms on the right-hand sides of Eqs (32) and
(33) represent the driving by the incident dominantmode fields in all the other waveguides of, respectively,
the dominant and higher-order modes in the a-wave46
The evaluation of the admittance matrix Ypo defined
in Eq.(10), and which is essential in the calculation of
the coupling admittances, (31), will in general require
a numerical computation. However, in the lower-hybrid
frequency range, an approximate decoupling of
Eqs (3-6) can be made. The result is an analytic
expression valid in most (but not all) of the Fourier
space (ny, n,).
For the fast wave, decoupling comes from assuming
a shorted parallel electric field, &z= 0, and the resulting
equation for Py is:
d2&gF
_ 2_ 2
+(
+iKi
n
K
nny-n
n 2-Ki
F=
(35)
so that we have: [&y, &z]F [F(),
01. For the slow
wave, we assume the local polarization, &y5 [nynz/
(n -K)]&z, and we find:
d2 .S
5 +
dW2
(nK - K
Ki-nK)nyPS=
)
(36)
with [Py,&zls= [&s,(nynz/(nz- K1 )) Ps]. Let us
assume a linear density profile near the plasma edge,
writing Kim 1, Ki= 1 -at, Kx= A where
NUCLEAR FUSION, Vol.23, No.1 (1983)
WAVEGUIDE-PLASMA COUPLING
d
dt
2
2
pe
d
t(=0
2
dt Pe,)/W2e) 6=0
and Eq.(36) can then be solved in terms of parabolic
cylinder functions and Airy functions, respectively.
The final result is:
,
+
Yo
igs(ny, nz)
igF(ny,nz)
f
iF(yInznynz(-
both cases, a TE mode is incident from the waveguide,
the difference being in the waveguide aperture's
orientation with respect to the static magnetic field.
The main result is that because, in general,
gS/gF ~ al/3 // 2 > 1, the incident polarization remains
dominant in the waveguide and most of the energy is
reflected in a TE mode.
For the slow-wave excitation, Einc c iEz and we can
estimate the resulting magnetic field at the plasma edge
from expression (37). The result is Hy/Hz-gs/gF)> I
and the polarization is approximately TE. The reflection coefficient is found as in Brambilla's theory, but
with an additional geometrical factor accounting for
the finite extent in y. Specifically, we calculate:
+ 2i)
YOo=
yz
-
zy (n-2]
(37)
fdnydnzigs(ny,nz)
OJ (27)2 (n2 -1)
(n
n )12
2
'
(40)
n-l
and the reflection coefficient is R =(Yeo- Y
where the admittance terms gs and gF are:
1/3 -iffI3
gs(nynz)=a
e
1/3
(n -l)
1/2
2,/4
/(Yw + YE)-
Ai'(0)
Ai(0)
(38)
For the fast-wave excitation we find a similar result.
The incident field is now E= fEy and the resulting
polarization from expression (37) has Hz- Hy. However a small Ez component
F(3/4 + a/2)
Ez/Ey - 0
and
(1-n
2
)2
__
(
1- n
with n z= ny + n . The expression (37) for
can
then be substituted into Eq.(31) for an evaluation of
the Ya. The limitations on the use of this approximation arise because of the resonant denominators
(nyz-- Iy' and (nZ-l)'. When Inyz-+1 or Inz-+l, the
assumptions which led to Eqs (35) and (36) break
down and Eqs (5) and (6) cannot be decoupled.
Furthermore, the resonant terms cannot simply be
ignored because they lead to a divergent integral in
expression (31). Thus, expression (37) is useful only
if the excitation spectrum has little energy near the
lines nz= 1 and the circle ny+ n2 = 1 in the (ny, n,)
space; then the singularities in the integral (31) can be
avoided by setting the spectrum to zero at some
arbitrary distance from the resonance. In effect, this
limits the application of expression (37) to phased
arrays which create a spectrum in nz which is well
above nz= ±1.
We now qualitatively consider excitations designed
to couple predominantly to slow or fast waves. In
NUCLEAR FUSION, vol.23, No.1 (1983)
will cancel the Hz, with almost all the energy reflected
in the TE mode. We find R = (Y00- Y4wE)/ oo+ YwE)
where:
Y
dnvZn
2 )2
(27)
igF(ny, nz)
2
-1
nyz-1'
IC
2
TE(ny,nz)j
(41)
In both cases, scattering into the other polarization
results in the excitation of a wave with amplitude
g13 2 /a 3 < 1 that of the dominant wave. The finite
extent in y results in the inclusion of the shape of the
ny spectrum in the -calculation of Y00(the lez,yTE(ny,nz)I
terms) and also possibly more evanescence (reactance)
for large values of ny.
7. SUMMARY AND DISCUSSION
We have derived a general formalism for the treatment of waveguide coupling in any frequency range,
provided the slab geometry adopted for the plasma
edge is valid. In particular, coupling in the ICRF and
ECRF frequency ranges as well as LHRF can be treated
47
BERS and THEILHABER
by this method. Furthermore, the formalism accounts
for finite waveguide dimensions both along and across
B0, a feature neglected in all previous treatments of the
problem, and allowance is made for the interaction of
all possible waveguide modes.
The plasma is characterized by four coupled firstorder differential equations which have to be solved for
a complete spectrum of wavenumbers. In certain
domains of wavenumber space, these equations can be
uncoupled and solved analytically. This is illustrated
in Section 6 for fast- and slow-wave coupling in the
LHRF. In general, however, solutions must be found
by numerical integration, and this was assumed in
Sections 4 and 5. We can proceed as follows. First,
we specify a density profile which joins smoothly to a
region of constant density far from- the waveguides.
In this region of constant density, plane waves propagate, and by choosing only outgoing waves we satisfy
the radiation conditions. Starting with arbitrary
amplitudes, we integrate back to the waveguides at
x = 0. This yields linear transfer matrices for eT(x)
and KT(x) in terms of the amplitudes of the plane
waves at large x. Eliminating these amplitudes yields
the admittance matrix Yp(x) defined in Eq.(7), and
with this admittance known, the problem is reduced to
solving a set of linear equations in the waveguide mode
amplitudes, as outlined in Sections 4 and 5, respectively,
Eqs (19) and (20) for a single waveguide and Eqs (32)
and (33) for multiple waveguides, i.e. waveguide arrays.
In general, the unknown waveguide mode amplitudes
consist of the reflected wave of the dominant mode
and all the higher-order modes. The coupling of these
amplitudes in the linear equations, just mentioned, is
given by expressions (18) and (31), respectively, for
the single waveguide and the waveguide arrays.
We note that the backward, numerical integration
of the plasma equations (3-6) assumes that the
resonance K1-O does not occur in the coupling region.
This is so in the LHRF where the LHR is chosen to be
in the central part of the plasma. However, in the
ICRF it may be that K1 -0 in the edge plasma region,
indicating that coupling can occur to the slow wave.
In such cases Eqs (3-6) may have to be supplemented
with thermal correction terms.
Finally, we should like to point out two limitations
of the model and analysis presented. The first relates
to the assumption that the plasma does not extend
into the waveguides. Waveguides containing a plasma
in a magnetic field can have mode structures [7] very
different from empty waveguides, and this may modify
the linear coupling problem in an important way.
Secondly, non-linear effects in the edge plasma, due
to pondermotive forces [9], can modify the plasma
dynamics and hence the coupling problem. Much
attention has been given to this recently for coupling
in the LHRF [10-13].
ACKNOWLEDGEMENTS
This work was supported in part by the US-DoE
Contract DE-AC02-78ET-51013, and in part by the
US-NSF Grant No.ENG 79-07047. The authors also
wish to thank J. Freidberg and D. Hewett for several
useful discussions.
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[1]
[2]
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[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
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(Manuscript received 17 November 1981
Final manuscript received 5 October 1982)
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NUCLEAR FUSION, Vol.23, No.1 (1983)